McLaughlin sporadic group
# McLaughlin sporadic group

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In the area of modern algebra known as group theory, the McLaughlin group McL is a sporadic simple group of order

    $2^7$ ⋅ $3^6$ ⋅ $5^3$ ⋅ 7 ⋅ 11 = 898,128,000
    ≈ $9×10^8.$

## History and properties

McL is one of the 26 sporadic groups and was discovered by Jack McLaughlin (1969) as an index 2 subgroup of a rank 3 permutation group acting on the McLaughlin graph with 275 = 1 + 112 + 162 vertices. It fixes a 2-2-3 triangle in the Leech lattice and thus is a subgroup of the Conway groups $\mathrm {Co} _{0}$, $\mathrm {Co} _{2}$, and $\mathrm {Co} _{3}$. Its Schur multiplier has order 3, and its outer automorphism group has order 2. The group 3.McL:2 is a maximal subgroup of the Lyons group.

McL has one conjugacy class of involution (element of order 2), whose centralizer is a maximal subgroup of type $2.A_8.$ This has a center of order 2; the quotient modulo the center is isomorphic to the alternating group $A_8.$

## Representations

In the Conway group $Co_3,$ McL has the normalizer McL:2, which is maximal in $Co_3.$

McL has 2 classes of maximal subgroups isomorphic to the Mathieu group $M_{22}.$ An outer automorphism interchanges the two classes of $M_{22}$ groups. This outer automorphism is realized on McL embedded as a subgroup of $Co_3.$

A convenient representation of $M_{22}$ is in permutation matrices on the last 22 coordinates; it fixes a 2-2-3 triangle with vertices the origin and the type 2 points x = (−3, $1^{23})$ and y = $(−4,-4,0^{22})'.$ The triangle's edge x-y = (1, 5, $1^{22})$ is type 3; it is fixed by a $Co_3.$ This $M_{22}$ is the monomial, and a maximal, subgroup of a representation of McL.

Wilson (2009) (p. 207) shows that the subgroup McL is well-defined. In the Leech lattice, suppose a type 3 point v is fixed by an instance of $\mathrm {Co} _{3}$. Count the type 2 points w such that the inner product v·w = 3 (and thus v-w is type 2). He shows their number is 552 = $2^3⋅3⋅23$ and that this $Co_3$ is transitive on these w.

|McL| = |Co3|/552 = 898,128,000.

McL is the only sporadic group to admit irreducible representations of quaternionic type. It has 2 such representations, one of dimension 3520 and one of dimension 4752.

## Maximal subgroups

Finkelstein (1973) found the 12 conjugacy classes of maximal subgroups of McL as follows:

* $U_4(3)$ order 3,265,920 index 275 – point stabilizer of its action on the McLaughlin graph
* $M_{22}$ order 443,520 index 2,025 (two classes, fused under an outer automorphism)
* $U_3(5)$ order 126,000 index 7,128
* $3^{1+4}:2.S_5$ order 58,320 index 15,400
* $3^4:M_{10}$ order 58,320 index 15,400
* $L_3(4):2_2$ order 40,320 index 22,275
* $2.A_8$ order 40,320 index 22,275 – centralizer of involution
* $2^4:A_7$ order 40,320 index 22,275 (two classes, fused under an outer automorphism)
* $M_{11}$ order 7,920 index 113,400
* $5+^{1+2}:3:8$ order 3,000 index 299,376

## Conjugacy classes

Traces of matrices in a standard 24-dimensional representation of McL are shown. The names of conjugacy classes are taken from the Atlas of Finite Group Representations.

Cycle structures in the rank 3 permutation representation, degree 275, of McL are shown.

| Class | Centraliser order | No. elements | Trace | Cycle type | |
| - | - | - | - | - | - |
| 1A | 898,128,000 | 1 | 24 |
| 2A | 40,320 | $3^4$ ⋅ $5^2$ ⋅ 11 | 8 | $1^{35},$ $2^{120}$ |
| 3A | 29,160 | $2^4$ ⋅ $5^2$ ⋅ 7 ⋅ 11 | -3 | $1^5,$ $3^{90}$ |
| 3B | 972 | $2^3$ ⋅ 3 ⋅ $5^3$ ⋅ 7 ⋅ 11 | 6 | $1^{14},$ $3^{87}$ |
| 4A | 96 | $2^2$ ⋅ $3^5$ ⋅ $5^3$ ⋅ 7 ⋅ 11 | 4 | $1^7,$ $2^{14},$ $4^{60}$ |
| 5A | 750 | $2^6$ ⋅ $3^5$ ⋅ ⋅ 7 ⋅ 11 | -1 | $5^{55}$ |
| 5B | 25 | $2^7$ ⋅ $3^6$ ⋅ 5 ⋅ 7 ⋅ 11 | 4 | $1^5,$ $5^{54}$ |
| 6A | 360 | $2^4$ ⋅ $3^4$ ⋅ $5^2$ ⋅ 7 ⋅ 11 | 5 | $1^5,$ $3^{10},$ $6^{40}$ |
| 6B | 36 | $2^5$ ⋅ $3^4$ ⋅ $5^3$ ⋅ 7 ⋅ 11 | 2 | $1^2,$ $2^6,$ $3^{11},$ $6^{38}$ |
| 7A | 14 | $2^6$ ⋅ $3^6$ ⋅ $5^3$ ⋅ 11 | 3 | $1^2,$ $7^{39}$ | power equivalent |
| 7B | 14 | $2^6$ ⋅ $3^6$ ⋅ $5^3$ ⋅ 11 | 3 | $1^2,$ $7^{39}$ | power equivalent |
| 8A | 8 | $2^4$ ⋅ $3^6$ ⋅ $5^3$ ⋅ 7 ⋅ 11 | 2 | 1, $2^3,$ $4^7,$ $8^{30}$ |
| 9A | 27 | $2^7$ ⋅ $3^3$ ⋅ $5^3$ ⋅ 7 ⋅ 11 | 3 | $1^2,$ 3, $9^{30}$ | power equivalent |
| 9B | 27 | $2^7$ ⋅ $3^3$ ⋅ $5^3$ ⋅ 7 ⋅ 11 | 3 | $1^2,$ 3, $9^{30}$ | power equivalent |
| 10A | 10 | $2^6$ ⋅ $3^5$ ⋅ $5^3$ ⋅ 7 ⋅ 11 | 3 | $5^7,$ $10^{24}$ |
| 11A | 11 | $2^7$ ⋅ $3^6$ ⋅ $5^3$ ⋅ 7 | 2 | $11^{25}$ | power equivalent |
| 11B | 11 | $2^7$ ⋅ $3^6$ ⋅ $5^3$ ⋅ 7 | 2 | $11^{25}$ | power equivalent |
| 12A | 12 | $2^5$ ⋅ $3^5$ ⋅ $5^3$ ⋅ 7 ⋅ 11 | 1 | 1, $2^2,$ $3^2,$ $6^4,$ $12^{20}$ |
| 14A | 14 | $2^6$ ⋅ $3^6$ ⋅ $5^3$ ⋅ 11 | 1 | 2, $7^5,$ $14^{17}$ | power equivalent |
| 14B | 14 | $2^6$ ⋅ $3^6$ ⋅ $5^3$ ⋅ 11 | 1 | 2, $7^5,$ $14^{17}$ | power equivalent |
| 15A | 30 | $2^6$ ⋅ $3^5$ ⋅ $5^2$ ⋅ 7 ⋅ 11 | 2 | 5, $15^{18}$ | power equivalent |
| 15B | 30 | $2^6$ ⋅ $3^5$ ⋅ $5^2$ ⋅ 7 ⋅ 11 | 2 | 5, $15^{18}$ | power equivalent |
| 30A | 30 | $2^6$ ⋅ $3^5$ ⋅ $5^2$ ⋅ 7 ⋅ 11 | 0 | 5, $15^2,$ $30^8$ | power equivalent |
| 30B | 30 | $2^6$ ⋅ $3^5$ ⋅ $5^2$ ⋅ 7 ⋅ 11 | 0 | 5, $15^2,$ $30^8$ | power equivalent |

## Generalized Monstrous Moonshine

Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For the Conway groups, the relevant McKay–Thompson series is $T_{2A}(\tau )$ and $T_{4A}(\tau )$.

